Robots and Taxes

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# Robots and Taxes
## EC 350: Labor Economics
### <a href="https://kyleraze.com">Kyle Raze</a>
### Winter 2022

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# Agenda

1. Factor demand
    - Hiring in the short run (review)
    - Hiring in the long run
    - Labor demand curve
3. Robot tax, featuring Bill Gates

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# Factor demand

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# Short run *vs.* long run

## **Short run**

> The time span over which a business can adjust some inputs (*e.g.,* labor), but cannot adjust others (*e.g.,* capital).

In the short run, we will assume that the level of employment **E** can vary, but capital **K** is fixed at an initial level **K.sub[0]**.

- **Example:** A shop foreman can hire or fire workers or adjust hours, but they are unable to expand the factory by adding assembly lines, heavy machinery, or a new building.

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# Short run *vs.* long run

## **Long run**

> The time span over which a business can adjust all inputs.

In the long-run, we will assume that both the level of employment **E** and capital **K** can vary.

- **Example:** An office manager can hire or fire workers, adjust hours, buy or sell desks and computers, or lease new office space.

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# Hiring in the short run

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## **Profit maximization**

An employer maximizes profit by hiring `$E^*$` workers where `$w = \text{MRP}_E$` and `$\text{MRP}_E$` is decreasing.

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# Hiring in the short run

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## .hi-purple[Labor demand]

The portion of the MRP curve below the VAP curve traces out the .hi-purple[short-run labor demand] curve.

- Describes how an employer adjusts employment as the market wage changes, holding other inputs constant.
- **Downward sloping:** An employer wants to reduce staffing as the wage increases, *all else equal*.

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# Hiring in the long run

## **Profit maximization**

In the long run, employers have the flexibility to adjust both labor and capital.

**Q:** How does the intuition for long-run factor demand compare to the intuition for the short run?

**A:** Employers will still make decisions ***at the margin*!**

- The underlying model is more elaborate and **our time is scarce**, so we will forgo a full derivation of the long-run profit maximization conditions.

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# Hiring in the long run

## **Profit maximization**

As in the short run, an employer will choose the profit-maximizing level of employment `$E^*$` such that `$$\begin{align} w &= \text{MRP}_E \\ &= p \times \text{MP}_E \end{align}$$`

- The employer will keep hiring until the **marginal cost** of the last worker **equals** the **marginal benefit** of the last worker.

Likewise, an employer will choose the profit-maximizing quantity of capital `$K^*$` such that `$$\begin{align} r &= \text{MRP}_K \\ &= p \times \text{MP}_K \end{align}$$`

- The employer will keep purchasing capital until the **marginal cost** of the last unit of capital **equals** the **marginal benefit** of the last unit of capital.

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# Hiring in the long run

## **Intuition?**

At the optimal, profit-maximizing bundle of labor and capital, we have `$$\begin{align} w = p \times \text{MP}_E && r = p \times \text{MP}_K \end{align}$$`

Dividing by marginal product, we obtain `$$\begin{align} p = \dfrac{w}{\text{MP}_E} && p = \dfrac{r}{\text{MP}_K} \end{align}$$`

Now we can equate both conditions to obtain the **long-run profit maximization condition** `$$\begin{align} \dfrac{w}{\text{MP}_E} = \dfrac{r}{\text{MP}_K} \end{align}$$`

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# Hiring in the long run

## **Intuition?**

**Q:** What can we learn from the long-run profit maximization condition? `$$\begin{align} \dfrac{w}{\text{MP}_E} = \dfrac{r}{\text{MP}_K} \end{align}$$`

- `$\dfrac{w}{\text{MP}_E}$` represents the cost of producing one more unit of output using labor.
- `$\dfrac{r}{\text{MP}_K}$` represents the cost of producing one more unit of output using capital.

**A:** At the profit-maximizing bundle, it would **not** be **cheaper to change** the mix of inputs.

- If it were cheaper to use relatively more labor than capital (or vice versa), then a **profit-maximizing employer would have already made the adjustment** (in the long run).

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# Hiring in the long run

## **Intuition?**

Suppose that `$w = 10$`, `$\text{MP}_E = 15$`, `$r = 5$`, and `$\text{MP}_K = 10$`. Then `$$\dfrac{w}{\text{MP}_E} = \dfrac{r}{\text{MP}_K} \implies \dfrac{10}{15} = \dfrac{5}{10} \implies \dfrac{2}{3} \neq \dfrac{1}{2}$$`

This employer is not profit maximizing!

- The cost of producing one more unit of output using labor exceeds the cost of producing one more unit using capital.
- It would be **more profitable** to use (relatively) **more capital** or **less labor**!

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# Hiring in the long run

**Q:** How does an employer respond to an increase in the market wage?

- **A.sub[1]:** The employer will respond by hiring fewer workers.

- **A.sub[2]:** The employer will adjust the level of capital, but the direction is theoretically ambiguous.
    - When the number of workers decreases, there are fewer people on each machine, which can reduce `$\text{MP}_K$`.
    - The direction of the response will depends on the **scale** and **substitution** effects.

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# Hiring in the long run

## **Scale effect**

Other things being equal, a **decrease** in the **price** of an input will **increase** the **quantity demanded** of that input.

- If the cost of production decreases, the employer will want to "scale up" production of the output good.
- Conversely, if the cost of production increases, the employer will "scale back" production.
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- Analogous to the wealth effect for a worker..super[.hi-pink[<span>&#8224;</span>]]

.footnote[.super[.hi-pink[<span>&#8224;</span>]] We assume that labor and capital are "normal" inputs&mdash;production increases as the amount of labor and capital increase.]

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# Hiring in the long run

## **Substitution effect**

Other things being equal, if the price of an input increases, **demand for the other input increases.**

- If labor becomes **relatively more expensive** than capital, then the employer will want to **substitute away** from labor and toward capital.
- If labor becomes **relatively cheaper** than capital, then the employer will want to **substitute toward** labor and away from capital.
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- Analogous to the substitution effect for the worker.

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# Hiring in the long run

## **Scale and substitution effects**

**Q:** How would a employer respond to an increase in the market wage?

|                     | Scale effect | Substitution effect |
|:-------------------:|:------------:|:-------------------:|
| `$\Delta$` in labor   | `$-$`          | `$-$`                 |
| `$\Delta$` in capital | `$-$`          | `$+$`                 |

**A.sub[K]:** For capital, it depends.

- If the scale effect dominates the substitution effect, then capital will eventually decrease.

**A.sub[E]:** For labor, the effect is unambiguous.

- The scale effect and substitution effect will move in the same direction for the input that undergoes a change in price.

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# Hiring in the long run

## **Scale and substitution effects**

**Q:** What determines whether the scale or substitution effect dominates?

**A:** Whether labor and capital are .hi-pink[substitutes] or .hi-purple[complements].

- .hi-pink[Substitutes:] Inputs used in place of one another.
    - Self-checkout kiosk *vs.* cashier
    - Tax prep software *vs.* accountant
    - Robot *vs.* low-skill worker?
- .hi-purple[Complements:] Inputs used together.
    - Carpenter and hammer
    - Mail carrier and mail truck
    - Robot and high-skill worker?

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# Hiring in the long run

### .hi-pink[Substitutes]

> Inputs used in place of one another.

Two inputs are said to be substitutes if the price of one input changes the demand of the other input .hi-pink[in the same direction.]

- Substitution effect outweighs the scale effect.

### .hi-purple[Complements]

> Inputs used together.

Two inputs are said to be complements if the price of one input changes the demand of the other input .hi-purple[in the opposite direction.]

- Scale effect outweighs the substitution effect.

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# Hiring in the long run

### **Cross-elasticity of factor demand**

> A unit-free measure of the responsiveness of demand for one input to a change in the price of the other.

**Labor responsiveness** to a change in the price of capital: `$$\eta = \dfrac{\%\Delta E}{\%\Delta r} = \dfrac{(E_2 - E_1) / E_1}{(r_2 - r_1) / r_1}$$`

**Capital responsiveness** to a change in the price of labor: `$$\eta = \dfrac{\%\Delta K}{\%\Delta w} = \dfrac{(K_2 - K_1) / K_1}{(w_2 - w_1) / w_1}$$`

`$\eta > 0$` .mono[-->] substitutes .mono[-->] substitution effect dominates. <br>
`$\eta < 0$` .mono[-->] complements .mono[-->] scale effect dominates.

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# Hiring in the long run

### **Cross-elasticity of factor demand**

> A unit-free measure of the responsiveness of demand for one input to a change in the price of the other.

**Q:** What is the cross-elasticity of factor demand implied by the data below?

|  Month   | Rental rate of capital | Hourly wage | Units of capital | Hours of labor |
|:--------:|:----------------------:|:-----------:|:----------------:|:--------------:|
| January  | 20                     | 15          | 5                | 60             |
| February | 25                     | 15          | 1                | 80             |

**A:** Roughly 1.33.

`$$\eta = \dfrac{\%\Delta E}{\%\Delta r} = \dfrac{(E_2 - E_1) / E_1}{(r_2 - r_1) / r_1} = \dfrac{(80-60)/60}{(25-20)/20} = \dfrac{1/3}{1/4} = \dfrac{4}{3} \approx 1.33$$`

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# Hiring in the long run

### **Cross-elasticity of factor demand**

> A unit-free measure of the responsiveness of demand for one input to a change in the price of the other.

**Q:** How do we interpret the cross-elasticity of factor demand we just calculated? What does 1.33 tell us?

**A.sub[1]:** A 1-percent increase in the price of capital .mono[-->] 1.33-percent increase in labor demanded.

**A.sub[2]:** Labor and capital are substitutes.

- Substitution effect .mono[>] scale effect

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# Market labor demand curve

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**Q:** Why is the short-run labor demand curve (.hi-purple[D.sub[SR]]) steeper than the short-run labor demand curve (.hi-purple[D.sub[LR]])?

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# Market labor demand curve

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**Q:** Why is the short-run labor demand curve (.hi-purple[D.sub[SR]]) steeper than the short-run labor demand curve (.hi-purple[D.sub[LR]])?

**A:** Firms have fewer alternatives to labor in the short run .mono[-->] less responsive to changes in the wage.
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# Responsiveness

### **Labor demand elasticity**

> A unit-free measure of the responsiveness of the quantity of labor demanded to changes in the wage.

`$$\epsilon_d = \dfrac{\%\Delta E}{\%\Delta w} = \dfrac{(E_2 - E_1) / E_1}{(w_2 - w_1) / w_1} \leq 0$$`

**Interpretation?** A 1-percent increase in wages .mono[-->] `$|\epsilon_d|$`-percent decrease in labor demanded.

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# Responsiveness

### **Labor demand elasticity**

> A unit-free measure of the responsiveness of the quantity of labor demanded to changes in the wage.

**Q:** What is the labor demand elasticity implied by the data below?

|  Month | Hourly wage | Hours of labor |
|:------:|:-----------:|:--------------:|
| June   | 15          | 120            | 
| July   | 20          | 60             |

**A:** -1.5.

`$$\epsilon_d = \dfrac{\%\Delta E}{\%\Delta w} = \dfrac{(E_2 - E_1) / E_1}{(w_2 - w_1) / w_1} = \dfrac{(60 - 120) / 120}{(20 - 15) / 15} = \dfrac{-1/2}{1/3} = -1.5$$`

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# Robot tax, featuring Bill Gates

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# Robot tax, featuring Bill Gates

## **Discussion**

**Q:** How would a robot tax affect...

- The relative price of robots (capital)?
- The employment of robots (capital) in the long run?
- The employment of labor in the long run?
    - Low-wage labor (*e.g.,* fast food workers, manual laborers, dishwashers, *etc.*)?
    - High-wage labor (*e.g.,* surgeons, engineers, software developers, *etc.*)?
- Tax revenue?

**Q:** Should we do it?

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# Housekeeping

**Problem Set 2** due by Saturday, February 5th by 11:59pm.

**Midterm exam** on Canvas on Monday, February 7th at 2:00pm.

- Extra office hours:
    - Friday from 3:00 to 5:00pm in PLC 522
    - Saturday from 12:00 to 2:00pm on Zoom.
- Study materials (all on Canvas):
    - **Midterm Review Guide**  
    - **Midterm Practice Problems** (w/ key)
    - Key to **Problem Set 1**
    - I will post a key to **Problem Set 2** after the due date